# Order of operations, precedence in Mathematica [duplicate]

I get confused in the order of operations in Mathematica. For example,

f /@ 10^{1, 2, 3}
{10, 100, 1000}


In my head, that should be

{f, f, f}


After using FullForm, I see that my command should be regarded as : (f /@ 10)^{1, 2, 3}. In fact, () is not often used in Mathematica code, and I'm always not sure about the order of the expression evaluation. And I can't find the information about this order.

For example, in C: http://en.wikipedia.org/wiki/Order_of_operations

On MathWorld, it is just the basic : http://mathworld.wolfram.com/Precedence.html

1. Parenthesization,

2. Factorial,

3. Exponentiation,

4. Multiplication and division,

Another example: let look at the order of this:

x + x /. x -> y
2 y

x + (x /. x -> y)
x + y


And it takes time to understand this:

x + y /. x -> 1 + y + x /. x -> 5
6 + 2 y


Without the order of operations, one can get confused by interpret this expression, and of course, we can use ().

((x + y) /. x -> 1) +(( y + x) /. x -> 5)
x + y /. x -> (1 + y + x) /. x -> 5


Another example, one can get confused:

Cases[{1, 2, 3}, _?#1 < 2 &]


or

Cases[{1, 2, 3}, _?(#1 < 2) &]


or

Cases[{1, 2, 3}, _?(#1 < 2 &)]


Another examples, which one & or /. is more privileged ?

x /. x -> y + #1 &

x + #1 & /. x -> y


I really appreciate any rule of thumbs or a guide of the operation ordered: /@, @@, _?, /., ->,& ...

• see "when is f@g not the same as f[g]?" mathematica.stackexchange.com/questions/30425/… Operator Precedence Table is posted there by MrWizard and more discussion on the topic Jan 20, 2014 at 1:30
• Thank you ! That's what I want to find. Jan 20, 2014 at 1:32

You can check the precedence using Precedence. So for you first example it works that way because Precedence[Power] < Precedence[Map]. You have furthermore that Precedence[Function] < Precedence[ReplaceAll] for one of your later queries.

When you get confused and need to tell what the precedence is the fastest way though is to use FullForm and Hold. There are variations of this, I forget which one is considered best practice, but it works like this:

FullForm[Hold[f /@ 10^{1, 2, 3}]]
(* Out: Hold[Power[Map[f,10],List[1,2,3]]] *)

• Be careful with Precedence, see my answer in the linked question.
– Kuba
Jan 20, 2014 at 7:07
• @Kuba I didn't read all of the comments carefully but it looks like nit-picking to me. The basic operation of Precedence appears to be solid. Jan 20, 2014 at 7:16
• @I mean the answer, not a comment. What I want to say is that Precedence@ReplaceAll does not differ if it is ReplaceAll[...] or .../.replacement and the syntax does matter, not necessarily in this case, but it did in my answer.
– Kuba
Jan 20, 2014 at 7:21